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1_channel.tex
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\documentclass[aspectratio=169]{beamer}
\def\stylepath{../styles}
\usepackage{\stylepath/com303}
\begin{document}
\begin{frame} \frametitle{digital communications}
\centering
\begin{figure}
\begin{dspBlocks}{1.2}{0.2}
\parbox{4ex}{\small \tt \bf ..01100\\ 01010...}\hspace{5ex} & \BDfilter{TX} & ~~~\parbox{8ex}{\small physical\\ channel} & \BDfilter{RX} & ~~~~~\parbox{4ex}{\small \tt \bf ..01100\\ 01010...} \\ \\
digital & & analog & & digital
\ncline{->}{1,1}{1,2}
\ncline{->}{1,2}{1,3}^{$s(t)$}
\ncline{->}{1,3}{1,4}^{$\hat{s}(t)$}
\ncline{->}{1,4}{1,5}
\end{dspBlocks}
\end{figure}
\end{frame}
%\begin{frame}
% \frametitle{a comparison of data rates}
% \begin{itemize}
% \item Transatlantic cable:
% \begin{itemize}
% \item 1866: 8 words per minute ($\approx$5 bps)
% \item 1956: AT\&T, coax, 48 voice channels ($\approx$3Mbps)
% \item 2005: Alcatel Tera10, fiber, 8.4 Tbps ($8.4\times 10^{12}$ bps)
% \item 2012: fiber, 60 Tbps
% \end{itemize}
% \pause
% \item Voiceband modems
% \begin{itemize}
% \item 1950s: Bell 202, 1200 bps
% \item 1990s: V90, 56Kbps
% \item 2008: ADSL2+, 24Mbps
% \end{itemize}
% \end{itemize}
%\end{frame}
%
%
%\begin{frame} \frametitle{Success factors for digital communications}
% \begin{enumerate}
% \item power of the digital paradigm
% \item natural integration with information theory
% \item progress in hardware components
% \end{enumerate}
%\end{frame}
%
%
%
%\begin{frame}
% \frametitle{Success factors for digital communications}
% 1) power of the DSP paradigm:
% \begin{itemize}
% \item integers are ``easy'' to regenerate
% \item good phase control in digital filters
% \item adaptive algorithms (equalization, echo cancellation)
% \end{itemize}
%\end{frame}
\def\binFun{x 23 div cvi 2 mod -2 mul 1 add 5 mul }
\def\fourFun{x 23 div cvi 4 mod -4 mul 2 add 5 mul }
\def\atten{10 div }
\def\noise{rand 2147483647 div 0.5 sub 0.2 mul add }
\def\ampli{10 mul }
%\begin{frame} \frametitle{Regenerating signals}
% \begin{center}
% \begin{figure}
% \begin{dspPlot}[yticks=none,xticks=none,sidegap=0]{0,99}{-7,7}
% \moocStyle
% \only<1|handout:1>{\dspFunc{\binFun}}
% \only<2|handout:2>{\dspFunc{\binFun \atten \noise}}
% \only<3|handout:3>{\dspFunc{\binFun \atten \noise \ampli}}
% \only<4|handout:4>{\dspFunc{\binFun}}
% \end{dspPlot}
% \end{figure}
% \only<1|handout:1>{$x(t)$}
% \only<2|handout:2>{$x(t)/G + \sigma(t)$}
% \only<3|handout:3>{$G[x(t)/G + \sigma(t)] = x(t) + G\sigma(t)$}
% \only<4|handout:4>{$\hat{x}_1(t) = G\mbox{sgn}[x(t) + \sigma(t)]$}
% \end{center}
%\end{frame}
%
%
%\begin{frame}
% \frametitle{Success factors for digital communications}
% 2) algorithmic nature of DSP is a perfect match with information theory:
% \begin{itemize}
% \item error correction (CD's and DVD's)
%% \item trellis-coded modulation and Viterbi decoding
% \item entropy coding (JPEG)
% \end{itemize}
%\end{frame}
%
%
%\begin{frame}
% \frametitle{Success factors for digital communications}
% 3) hardware advancement
% \begin{itemize}
% \item general-purpose platforms
% \item miniaturization
% \item power efficiency
% \end{itemize}
%\end{frame}
\note{\vspace{10em} Introduce the notion of physical channel and its many forms. Each channel type will have different specs and will need different strategies}
\def\cphone{\raisebox{-1.2em}{ \includegraphics[height=4em]{iphone}}}
\def\phone{\raisebox{-1.2em}{ \includegraphics[height=4em]{phone.eps}}}
\begin{frame} \frametitle{The many incarnations of a conversation}
\begin{figure}
\psset{linearc=0.2}
\begin{dspBlocks}{1.2}{0.1}
\cphone & \BDfilter{Base Station} & [name=A]\BDfilter{Switch} & \\
& & & [mnode=circle,name=B] Network \\
\phone & \BDfilter{Switch} & [name=C]\BDfilter{CO} &
\ncline[linestyle=dotted]{->}{1,1}{1,2}\taput{air~~~~~~}
\ncline{->}{1,2}{1,3}\taput{~~~copper}
\ncline{->}{3,3}{3,2}\taput{~~~~~coax}
\ncline{->}{3,2}{3,1}\taput{~~copper}
\end{dspBlocks}
\ncdiagg[angleA=0,angleB=90,arm=1cm,linewidth=1.2pt]{->}{A}{B}
\taput{~~~fiber}
\ncdiagg[angleA=0,angleB=90,arm=1cm,linewidth=1.2pt]{<-}{C}{B}
\end{figure}
\end{frame}
\begin{frame} \frametitle{The analog channel}
the constraints of all physical channels:
\begin{itemize}
\item bandwidth constraint
\item power constraint
\end{itemize}
\vspace{1em}
\centering
{both constraints will affect the final {\em capacity} of the channel}
\end{frame}
\begin{frame} \frametitle{Channel constraints}
\centering
baseband channel
\begin{dspPlot}[height=2cm,xticks=custom,yticks=none,sidegap=0]{-7,7}{0,1.1}
\moocStyle
\psframe[fillstyle=vlines,linecolor=green!70,hatchcolor=green!70,hatchangle=20](-2,0)(2,0.5)
\dspCustomTicks[axis=x]{0 0 2 $f_{\max}$}
\psset{linecolor=black}
\psbrace[rot=-90,ref=tC,nodesepB=-8pt](2,.53)(-2,.53){max bandwidth $W=2f_{\max}$}
\psbrace[ref=lC,nodesepA=5pt](2.25,0)(2.25,.5){area is max power}
\end{dspPlot}
\vspace{1em}
passband channel
\begin{dspPlot}[height=2cm,xticks=custom,yticks=none,sidegap=0]{-7,7}{0,1.1}
\moocStyle
\psframe[fillstyle=vlines,linecolor=green!70,hatchcolor=green!70,hatchangle=20](1,0)(5,0.25)
\psframe[fillstyle=vlines,linecolor=green!70,hatchcolor=green!70,hatchangle=20](-1,0)(-5,0.25)
\dspCustomTicks[axis=x]{0 0 3 $f_{c}$ 5 $f_c+W/2$}
\psset{linecolor=black}
\psbrace[rot=-90,ref=tC,nodesepB=-8pt](5,.253)(1,.253){max bandwidth $W$}
\psbrace[ref=lC,nodesepA=5pt](5.25,0)(5.25,.25){area is max power}
\end{dspPlot}
\end{frame}
\begin{frame} \frametitle{Example: the AM radio channel}
\centering
\begin{figure}[t]
\begin{dspBlocks}{1.3}{0.4}
& & $\cos(2\pi f_c t)$ &
{%
\psset{xunit=1em,yunit=1em}%
\pspicture(-2,-2)(2,0)%
\psline[linewidth=1pt](0,-2)(-1.5,0)(1.5,0)(0,-2)
\endpspicture}\\
$x(t)$~ & \BDlowpass & \BDmul &
\ncline{->}{1,3}{2,3}
\ncline{2,1}{2,2}
\ncline{->}{2,2}{2,3}
\ncline{2,3}{2,4}
\ncline{2,4}{1,4}
\end{dspBlocks}
\end{figure}
\vspace{2em}
block diagram of a simple AM trasmitter
\end{frame}
\begin{frame} \frametitle{Example: the AM radio channel}
\begin{itemize}
\item AM band is from 530kHz to 1.7MHz
\item each radio channel is 8KHz wide
\item power limited by law:
\begin{itemize}
\item daytime/nighttime
\item interference
\item health hazards
\end{itemize}
\end{itemize}
\end{frame}
\def\phone{\raisebox{-1.2em}{ \includegraphics[height=3em]{phone.eps}}}
\begin{frame} \frametitle{Example: the telephone channel}
\centering
\begin{figure}[t]
\psset{linearc=0.2,angleB=180}
\begin{dspBlocks}{0.8}{0.4}
& & & & & \phone \\
\phone & \BDfilter{CO} & [mnode=circle] Network & \BDfilter{CO} & \phone \\
& & & & & \phone
\end{dspBlocks}
\ncline{->}{2,1}{2,2}
\ncline{->}{2,2}{2,3}
\ncline{->}{2,3}{2,4}
\ncdiag[lineAngle=80]{->}{2,4}{1,6}
\ncline[linestyle=dotted]{->}{2,4}{2,5}
\ncdiag[lineAngle=-80,linestyle=dotted]{->}{2,4}{3,6}
\end{figure}
\end{frame}
\begin{frame} \frametitle{Example: the telephone channel}
\begin{itemize}
\item from around 300Hz to around 3500Hz
\item power limited by law to 0.2-0.7V rms
\item noise is rather low: SNR usually 30dB or more
\end{itemize}
\end{frame}
\begin{frame} \frametitle{Example: optical fiber}
\begin{itemize}
\item many types of fiber, eg. multi-mode or single-mode
\item MMF at 850nm typically 500 MHz/km
\item SMF at 1300nm has practically infinite bandwidth
\item power limited by fiber size, a few hundred mW
\end{itemize}
\centering
\includegraphics[height=5cm]{fiber.eps}
\end{frame}
\begin{frame} \frametitle{Channel capacity}
\note<1>{\vspace{10em} information and reliability are fuzzy concepts at this time but they will be clearer as we go along. stress the intuitive part of both}
\centering
maximum amount of information that can be {\em reliably} delivered over the channel \\
(bits per second)
\end{frame}
\begin{frame} \frametitle{About reliability}
\centering
we cannot design a perfect (error-free) communication system because of noise
\vspace{1em}
but
\vspace{1em}
we can design a system with arbitrary small error rate (e.g. $10^{-6}$)
\end{frame}
\begin{frame} \frametitle{Capacity formula for the Gaussian channel}
\begin{center}
\[
C = W \log_2(1+ \mbox{SNR}) \quad \mbox{bits/s}
\]
\vspace{3em}
\it (Shannon, 1949)
\end{center}
\end{frame}
\begin{frame} \frametitle{chasing capacity}
\centering
\begin{dspBlocks}{1.2}{1.2}
$a[n]$ & \BDfilter{TX} & \BDadd & \BDfilter{RX} & ~~~$\hat{a}[n]$ \\
& & noise
\ncline{->}{1,1}{1,2}
\ncline{->}{1,2}{1,3}^{$s(t)$}
\ncline{->}{1,3}{1,4}^{$\hat{s}(t)$}
\ncline{->}{1,4}{1,5}
\ncline{->}{2,3}{1,3}
\end{dspBlocks}
\vspace{1em}
\begin{itemize}
\item $a[n]$: stream of digital samples at $B$ samples per second
\item $s(t)$: transmitted analog signal fullfilling channel constraints
\item $\hat{s}(t)$: received noisy analog signal
\item $\hat{a}[n]$: decoded samples (with possible errors)
\end{itemize}
\end{frame}
\begin{frame} \frametitle{The design problem}
\begin{itemize}
\item samples $a[n]$ drawn from a finite set of symbols $\mathcal{A}$
\item $R = \log_2(|\mathcal{A}|)$: bits per symbol
\item $B$: number of symbols (samples) per second
\end{itemize}
\vspace{2em}
\begin{center}
bit rate: $BR$ bits per second
\end{center}
\end{frame}
\begin{frame} \frametitle{Bandwidth vs capacity: intuition}
\begin{itemize}
\item to transmit $a[n]$ over an analog channel, sinc-interpolate $a[n]$ with a period $T_s$
\item if we make $T_s$ small we can send more symbols per unit of time... \\
... but the bandwidth of the signal will grow as $1/T_s$
\end{itemize}
\vspace{1em}
\centering
the maximum symbol rate is equal to the bandwidth constraint, $B = W$
\vspace{2em}
\it (Nyquist, 1924)
\end{frame}
\begin{frame} \frametitle{Sinc interpolation}
\begin{align*}
x(t) &= \sum_{n = -\infty}^{\infty}x[n]\,\mbox{sinc}\left(\frac{t - nT_s}{T_s}\right) \\ \\
X(f) &= \begin{cases}
\frac{1}{F_s}X(e^{j2\pi f/F_s}) & \mbox{for $|f| \leq F_s/2 = \displaystyle\frac{1}{2T_s}$} \\
0 & \mbox{otherwise}
\end{cases}
\end{align*}
\end{frame}
\def\plotSpec#1#2{\dspFunc[#2]{x \dspPorkpie{0}{#1} 2 mul}} % #1 div}}
\def\plotCurrent#1#2#3#4#5{%
\only<#1>{
\FPupn\o{#2 1 / 2 trunc}
\plotSpec{\o}{}
\dspCustomTicks[axis=x]{0 0 {-\o} #5 {\o} #4}
\dspText(-2,1.8){$T_s=$#3}}}
\def\plotPast#1#2{
\only<#1->{
\FPupn\o{#2 1 / 3 trunc}
\plotSpec{\o}{linecolor=gray}}}
\begin{frame} \frametitle{Spectrum of interpolated signals}
\centering
\begin{figure}
\begin{dspPlot}[xtype=freq,height=2cm,ylabel={$X(e^{j\omega})$}]{-1,1}{0,1.2}
\moocStyle
\dspFunc{x \dspPorkpie{0}{1}}
\end{dspPlot}
\begin{dspPlot}[xtype=freq,xticks=custom,yticks=custom,height=2cm,ylabel={$X(f)$}]{-2.5,2.5}{0,2.4}
\moocStyle
\plotCurrent{1}{2}{$T_0$}{$1/(2T_0)$}{$-1/(2T_0)$}
\plotPast{2}{2}
\plotCurrent{2}{1}{$T_0/2$}{$1/T_0$}{-$1/T_0$}
\plotPast{3}{1}
\plotCurrent{3}{0.5}{$T_0/4$}{$2/T_0$}{$-2/T_0$}
\end{dspPlot}
\end{figure}
\end{frame}
\begin{frame} \frametitle{Power vs capacity: intuition (I)}
noise margin:
\begin{itemize}
\item $\mathcal{A} = \{a_1, \ldots, a_{N}\}$; assume $\max_k\{|a_k|\} = 1$
\item when transmitter sends $\alpha\in\mathcal{A}$, receiver gets noisy $\hat{\alpha} = \alpha + \eta$
\item error when $\min_{a\in\mathcal{A}}|\hat{\alpha} - a| \neq \alpha$
\item noise margin for $\mathcal{A}$:
\[
d_{\mathcal{A}} = \min_{0\le h < N}\min_{k\neq h}|a_h - a_k| / 2
\]
\item noise margin is a decreasing function of $N$; eg:
\[
\mathcal{A} = \{1/N, 2/N, \ldots, 1\} \qquad \Rightarrow \qquad d_{\mathcal{A}} = 1/(2N)
\]
\[
\mathcal{A} = \{-1, -1 + 2/N, \ldots, 1 -2/N, 1\} \qquad \Rightarrow \qquad d_{\mathcal{A}} = 1/N
\]
\end{itemize}
\end{frame}
\begin{frame} \frametitle{1 bit per symbol}
\begin{center}
\begin{figure}
\begin{dspPlot}[yticks=custom,xticks=custom,sidegap=0,xout=true]{0,99}{-7,7}
\moocStyle
\only<1>{\dspFunc{\binFun}}
\only<2>{
\dspFunc{\binFun \atten \noise \ampli}
\psline[linewidth=2.5pt,linecolor=blue]{<->}(10,0)(10,5)
\dspText(22,2){\color{blue} noise margin}}
\dspCustomTicks[axis=x]{23 $T_s$}
\dspCustomTicks[axis=y]{-5 $-1$ 5 $1$}
\end{dspPlot}
\end{figure}
\end{center}
\end{frame}
\begin{frame} \frametitle{2 bits per symbol}
\begin{center}
\begin{figure}
\begin{dspPlot}[yticks=custom,xticks=custom,sidegap=0,xout=true]{0,99}{-4,4}
\moocStyle
\only<1>{
\dspFuncOpt{/A [1 3 -1 -3 1] def}{x 23 div cvi A exch get}}
\only<2>{
\dspFuncOpt{/A [1 3 -1 -3 1] def}{x 23 div cvi A exch get \noise}
\psline[linewidth=2pt,linecolor=blue]{<->}(10,0)(10,1)
\dspText(22,0.3){\color{blue} noise margin}}
\dspCustomTicks[axis=x]{23 $T_s$}
\dspCustomTicks[axis=y]{-3 $-1$ 3 $1$}
\end{dspPlot}
\end{figure}
\end{center}
\end{frame}
\begin{frame} \frametitle{Power vs capacity: intuition (II)}
noise margin:
\begin{itemize}
\item transmission gain $G > 0$ scales the symbols in $\mathcal{A}$:
\[
\mathcal{A}_G = \{Ga_1, \ldots, Ga_{N}\}
\]
\item noise margin is linear: $d_{\mathcal{A}_G} = Gd_{\mathcal{A}}$
\item reliability is an \textit{increasing} function of noise margin
\item target reliability sets minimum noise margin: $d_{\mathcal{A}_G} \ge D$
\item target reliability sets minimum transmission gain for a given $\mathcal{A}$:
\[
G \ge D/d_{\mathcal{A}}
\]
\end{itemize}
\end{frame}
\begin{frame} \frametitle{Power vs capacity: intuition (III)}
$C=BR$ but $B_{\max} = W$; to increase $C$ we must increase $R = \log_2|\mathcal{A}| = \log_2 N$
\begin{itemize}
\item signal power $\sigma_a^2 = \expt{|a[n]|^2} \le G^2$
\item reliability requires $G \ge D/d_{\mathcal{A}}$
\item power constraint requires $G^2 \le \sigma_{\max}^2$
\item power constraint caps reliability: $D/d_{\mathcal{A}} \le \sigma_{\max}$
\item power constraint caps N: e.g. if $d_{\mathcal{A}} = 1/N$, then $N \le \sigma_{\max}/D$
\item the power constraint limits $R = \log_2 N$
\end{itemize}
\vspace{1em}
\begin{center}
\it (Hartley, 1928)
\end{center}
\end{frame}
\intertitle{Designing a digital transmitter}
\begin{frame}
\frametitle{The all-digital paradigm}
\centering
keep everything digital until we hit the physical channel
\vspace{3em}
\begin{figure}
\begin{dspBlocks}{1.8}{0.2}
\parbox{4ex}{\small \tt \bf ..01100\\ 01010...}\hspace{5ex} & \BDfilter{TX} & \BDfilter{D/A} & $~~s(t)$ \\
& & $F_s = 1/T_s$
\ncline{->}{1,1}{1,2}
\ncline{->}{1,2}{1,3}^{$s[n]$}
\ncline{->}{1,3}{1,4}
\end{dspBlocks}
\end{figure}
\end{frame}
\begin{frame} \frametitle{The channel constraints}
\begin{figure}
\begin{dspPlot}[height=3cm,xticks=custom,yticks=none,sidegap=0]{0,5}{0,1.1}
\moocStyle
\psframe[fillstyle=vlines,linecolor=green!70,hatchcolor=green!70,hatchangle=20](1,0)(2.4,0.5)
\dspCustomTicks[axis=x]{0 0 1 $F_{\min}$ 2.4 $F_{\max}$}
\psset{linecolor=black}
\psbrace[rot=-90,ref=tC,nodesepB=-15pt](2.4,.53)(1,.53){bandwidth constraint}
\psbrace[ref=lC,nodesepA=5pt](2.55,0)(2.55,.5){power constraint}
\end{dspPlot}
\end{figure}
\end{frame}
\begin{frame} \frametitle{Converting the specs to a digital design}
\begin{figure}
\begin{dspPlot}[height=3cm,xticks=custom,yticks=none,sidegap=0]{0,5}{0,1.1}
\moocStyle
\psframe[fillstyle=vlines,linecolor=green!70,hatchcolor=green!70,hatchangle=20](1,0)(2.4,0.5)
\dspCustomTicks[axis=x]{0 0 1 $F_{\min}$ 2.4 $F_{\max}$}
\only<2>{\dspCustomTicks[axis=x]{4 ${\color{red}F_s/2}$}}
\end{dspPlot}
\end{figure}
\end{frame}
\begin{frame} \frametitle{Converting the specs to a digital design}
\begin{figure}
\begin{dspPlot}[height=3cm,xticks=1,xtype=freq,yticks=none]{0,1}{0,1.1}
\moocStyle
\psframe[fillstyle=vlines,linecolor=green!70,hatchcolor=green!70,hatchangle=20](.25,0)(.6,0.5)
\dspCustomTicks[axis=x]{.25 $\omega_{\min}$ .6 $\omega_{\max}$}
\end{dspPlot}
\end{figure}
\end{frame}
\begin{frame} \frametitle{Transmitter design}
\intro{points to remark: this is independent on the structure of the symbols, on their distribution or values}
some working hypotheses:
\begin{itemize}
\item convert the bitstream into a sequence of symbols $a[n]$ via a mapper
\item model $a[n]$ as a white random sequence (add a scrambler on the bitstream to make sure)
\item now we need to convert $a[n]$ into a continuous-time signal within the constraints
\end{itemize}
\vspace{1em}
\begin{figure}
\psset{linearc=0.2}
\begin{dspBlocks}{1.0}{0.8}
\parbox{4ex}{\small \tt \bf ..01100\\ 01010...}\hspace{5ex} &
\BDfilter{Scrambler} & \BDfilter{Mapper} & \BDfilter{?} & $~~s(t)$\\
& &
\ncline{->}{1,1}{1,2} \ncline{->}{1,2}{1,3}
\ncline{->}{1,3}{1,4}^{~~~~$a[n]$}\ncline{->}{1,4}{1,5}
\end{dspBlocks}
\end{figure}
\end{frame}
\end{document}
\begin{figure}
\psset{linearc=0.2}
\begin{dspBlocks}{1.0}{0.8}
\parbox{4ex}{\small \tt \bf ..01100\\ 01010...}\hspace{5ex} &
\BDfilter{Scrambler} & \BDfilter{Mapper} & [name=A,mnode=circle] $K \uparrow$ \\
& &
\ncline{->}{1,1}{1,2} \ncline{->}{1,2}{1,3}
\ncline{->}{1,3}{1,4}^{~~~~$a[n]$}
\end{dspBlocks}
\begin{dspBlocks}{1.2}{0.4}
& [name=B] \BDfilter{$G(z)$} & \BDmul & \BDfilter{Re} & \BDfilter{$I(t)$} & $s(t)$ \\
& & $e^{j\omega_c n}$
\ncline{->}{1,2}{1,3}^{$b[n]$} \ncline{->}{1,3}{1,4}^{$c[n]$}
\ncline{->}{1,4}{1,5}^{$s[n]$} \ncline{->}{1,5}{1,6} \ncline{->}{1,6}{1,7}
\ncline{->}{2,3}{1,3}
\end{dspBlocks}
\ncbarr[angleA=0,arm=1cm,linewidth=1.2pt]{A}{B}
\end{figure}

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